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Fractal Geometry and Complex Dimensions in MetricMeasure Spaces
Sean Watson
University of California Riverside
June 14th, 2014
Sean Watson (UCR) Complex Dimensions in MM Spaces 1 / 56
Overview
1 Metric Measure SpacesExamples
2 Fractal Geometry and Complex DimensionsFractal StringsDistance Zeta FunctionTube Zeta Function
3 Generalizing the Theory to MM Spaces
Sean Watson (UCR) Complex Dimensions in MM Spaces 2 / 56
Main References
FGCD M.L. Lapidus, M. Frankenhuijsen: Fractal Geometry, ComplexDimensions and Zeta Functions: Geometry and Spectra of FractalStrings, Second Edition (of the 2006 First Edition), SpringerMonograph in Mathematics, Springer, New York, 2013, 593 pages.
FZF M.L. Lapidus, G. Radunovic, D. Zubrinic: Fractal Zeta Functions andFractal Drums: Higher-Dimensional Theory of Complex Dimensions,research monograph, preprint, 317 pages. Expected submission date:June 2014.
Sean Watson (UCR) Complex Dimensions in MM Spaces 3 / 56
Metric Measure Spaces
Metric Measure Space
A Metric Measure Space (or MM space) is a set X equipped with ametric d and a positive Borel measure µ that is doubling; ∃C positive suchthat
µ(Bd(x , 2r)) ≤ Cµ(Bd(x , r)).
Sean Watson (UCR) Complex Dimensions in MM Spaces 4 / 56
Metric Measure Spaces
Given a metric, there is always a measure that we can construct on thespace:
Hausdorff Measure
For any s nonnegative, A ⊂ X , define the s-dimensional Hausdorff outermeasure:
Hs(A) = limδ→0
Hsδ = lim
δ→0inf
{ ∞∑j=1
(diamEj)s : A ⊂
∞⋃j=1
Ej , diamEj < δ
}.
A set A has Hausdorff dimension DH if
DH = inf{s ≥ 0 : Hs(A) = 0}= sup{s ≥ 0 : Hs(A) =∞}.
We call HD the D-dimensional Hausdorff measure when restricted toBorel sets.
Sean Watson (UCR) Complex Dimensions in MM Spaces 5 / 56
Metric Measure Spaces
0
∞
Hs(A)
DH
s
An example of the s-dimensional Hausdorff measure of a set A withHausdorff dimension DH .
Sean Watson (UCR) Complex Dimensions in MM Spaces 6 / 56
Metric Measure Spaces
MM spaces are a growing area of research, especially in fields such as:
Harmonic Analysis
Partial Differential Equations
Probability Theory
Function Spaces
Geometric Analysis on Non-Smooth Spaces
Analysis on Fractals
Sean Watson (UCR) Complex Dimensions in MM Spaces 7 / 56
Metric Measure Spaces
We will see that we need a stronger requirement than just the doublingcondition:
Ahlfors regularity
A MM space is Ahlfors regular of dimension D (here on, regular) if∃K > 0 such that
K−1rD ≤ µ(B(x , r)) ≤ KrD
∀x ∈ X , 0 < r ≤ diamX .
If only the upper (resp. lower) bounds are satisfied, we call the spaceupper (resp. lower) Ahlfors regular of dimension D.
Sean Watson (UCR) Complex Dimensions in MM Spaces 8 / 56
Metric Measure Spaces
The measure µ of a regular D dimensional space and HD are equivalent inthat there is a constant C depending only on K such that
C−1µ(E ) ≤ HD(E ) ≤ Cµ(E ) ∀ Borel E ⊆ X .
In particular, if the MM space triple (X , d , µ) is regular of dimension D,then so is (X , d ,HD).
Sean Watson (UCR) Complex Dimensions in MM Spaces 9 / 56
Symbolic Cantor sets
Symbolic Cantor set
Let F be a finite set with k ≥ 2 elements. Then F∞ = {{xi}∞i=1 : xi ∈ F}is the k-Cantor set.
Define the valuation L(x , y), x = {xi}∞i=1, y = {yi}∞i=1, by
L(x , y) = `,
where xi = yi ∀i ≤ `, x`+1 6= y`+1.
Sean Watson (UCR) Complex Dimensions in MM Spaces 10 / 56
Symbolic Cantor sets
Let a ∈ (0, 1). Thenda(x , y) = aL(x ,y)
is an ultrametric.
Place the natural probability measure µ =∏∞
i=1 ν, ν(j) = 1/k for j ∈ F .
Then F∞ is regular of dimension D =log k
log a−1.
Sean Watson (UCR) Complex Dimensions in MM Spaces 11 / 56
Symbolic Cantor sets
The Cantor set is generated by successive removals of middle thirds.
Sean Watson (UCR) Complex Dimensions in MM Spaces 12 / 56
Laakso graph
Let X0 = [0, 1]. For i > 0, define Xi by replacing each edge of Xi−1 by a4−(i−1) scaled copy of Γ (below). Then {Xi}∞i=0 forms an inverse system
X0π0←− . . . πi−1←−− Xi
πi←− . . . ,
where πi−1 : Xi → Xi−1 collapses the copies of Γ at the i-th level.
Graph of Γ
Sean Watson (UCR) Complex Dimensions in MM Spaces 13 / 56
Laakso graph
Then the inverse limit X∞ is the Laakso graph, with metric
d∞(x , x ′) = limi→∞
dXi(π∞i (x), π∞i (x ′)),
where X∞ is the Gromov-Hausdorff limit of {Xi} and π∞i : X∞ → Xi isthe canonical projection.
The Laakso graph is a compact, and hence complete, metric space. Thisguarantees the existence of a doubling measure that makes the Laaksograph an MM space.
Of similar construction is the Laakso space, which is regular of dimension
D = 1 +log 2
log 3.
Sean Watson (UCR) Complex Dimensions in MM Spaces 14 / 56
Laakso graph
Sean Watson (UCR) Complex Dimensions in MM Spaces 15 / 56
Heisenberg Group
Heisenberg Group
We define the n-dimensional Heisenberg group in its algebrarepresentation Hn = Cn × R with group multiplication given by
(z , t)(z ′, t ′) = (z + z ′, t + t ′ − 1
2Im
n∑j=1
zjz ′j ).
There is a natural dilation action ∂r (z , t) = (rz , r2t), r > 0, that gives rise
to the homogeneous norm ‖(z , t)‖ = (∑n
j=1 |zj |4 + t2)14 , with the
properties ‖∂r (z , t)‖ = r‖(z , t)‖ and ‖x−1‖ = ‖x‖.
Sean Watson (UCR) Complex Dimensions in MM Spaces 16 / 56
Heisenberg Group
This norm defines a metric d(x , y) = ‖x−1y‖.
The Haar measure given by Lebesgue measure under the exponential mapgives
µ(B(x , r)) = r2n+2µ(B(x , 1)).
Thus the Heisenberg group is regular of dimension D = 2n + 2, while itstopological dimension is T = 2n + 1.
Sean Watson (UCR) Complex Dimensions in MM Spaces 17 / 56
Fractals and Harmonic embeddings
Many self-similar fractals in Euclidean space can be thought of as MM orAhlfors regular spaces.
Using key work of Kusuoka, Kigami showed that the Sierpinski gasketcould be embedded in R2 by a certain harmonic map. He also showed theresulting harmonic Sierpinski gasket can be viewed as a measurableRiemannian manifold with the intrinsic geodesic metric induced by theEuclidean structure of R2.
More recently, Kajino proved that the harmonic gasket is an Ahlforsregular space under the associated Hausdorff measure.
Sean Watson (UCR) Complex Dimensions in MM Spaces 18 / 56
Sierpinski Gasket
Sean Watson (UCR) Complex Dimensions in MM Spaces 19 / 56
Harmonic Sierpinski Gasket
Sean Watson (UCR) Complex Dimensions in MM Spaces 20 / 56
Weighted RN
Weighted RN
We define weighted RN space as the triple (RN , d , µ), where d is thestandard Euclidean metric and µ is the measure defined as
dµ = |x |αdx , α > −N, dx Lebesgue measure.
Then weighted RN space is a MM space, but it is not Ahlfors regular.
Sean Watson (UCR) Complex Dimensions in MM Spaces 21 / 56
What defines a fractal?
Two dimensions that capture how volume scales under dilations orcontractions: the Hausdorff dimension and the Minkowski dimension.
Minkowski dimension
Given A ⊂ RN bounded, define the t-neighborhood of A byAt := {x ∈ RN : d(x ,A) < t}. Then the r-dimensional Minkowskiupper content is defined as
M∗r = lim supt→0
|At |tN−r
.
Define the upper Minkowski dimension by
dimBA = inf{r ∈ R :M∗r (A) = 0}= sup{r ∈ R :M∗r (A) =∞}.
Sean Watson (UCR) Complex Dimensions in MM Spaces 22 / 56
What defines a fractal?
0
∞
r
M∗r
dimBA
An example of the r -dimensional Minkowski upper content of a set A withMinkowski dimension dimBA .
Sean Watson (UCR) Complex Dimensions in MM Spaces 23 / 56
What defines a fractal?
We define the r-dimensional lower Minkowski content Mr∗ and lower
Minkowski dimension dimB(A) analogously.
Minkowski dimension cont.
Given a set A ⊂ R, ifdimBA = dimBA,
then we call their common value dimBA the Minkowski dimension of A.
However, it is not enough to say that non-integer scaling dimensions orscaling dimensions larger than topological dimension defines a fractal.
Sean Watson (UCR) Complex Dimensions in MM Spaces 24 / 56
What defines a fractal?
The Devil’s staircase, which has Minkowski, Hausdorff and topologicaldimensions D = 1.
Sean Watson (UCR) Complex Dimensions in MM Spaces 25 / 56
What defines a fractal?
However, as done in [FGCD], we find that by studying the volume of theε-neigborhoods of the Devil’s staircase, it is approximated by
V (ε) ≈ 2ε2−1 +4− π8 log 3
∞∑n=−∞
ε(2−D−inp)
(D + inp)(1− D − inp),
where D = log3 2 and p = 2π/ log 3.
Viewing the exponents of ε as codimensions, we see that the dimensionsassociated to the geometry of the Devil’s staircase are 1 and D + inp forany n ∈ Z. In particular, there are complex values associated to thesedimensions. We call the entire set of dimensions the complex dimensionsof the Devil’s staircase.
Sean Watson (UCR) Complex Dimensions in MM Spaces 26 / 56
What defines a fractal?
p
0 D 1
The complex dimensions of the Devil’s staircase, with D = log3 2 andp = 2π/ log 3.
Sean Watson (UCR) Complex Dimensions in MM Spaces 27 / 56
Fractal Strings
The theory of complex dimensions in R was developed through the use offractal strings (one-dimensional fractal drums) in [FGCD].
Fractal String
A fractal string is a bounded open subset of the real line; i.e. it is adisjoint union of open intervals (the boundary of which may be fractal).The lengths of these open intervals forms a non-increasing sequence
L = `1, `2, `3, . . .
For example, we define the Cantor string, CS, as the complement of theternary Cantor set in [0, 1]. Thus in terms of the lengths of the disjointopen intervals,
CS =
{1
3,
1
9,
1
9,
1
27,
1
27,
1
27,
1
27, . . .
}
Sean Watson (UCR) Complex Dimensions in MM Spaces 28 / 56
Fractal Strings
1
9
1
3
ℓ1
1
9
1
27
1
27
1
27
1
27
ℓ2 ℓ3 ℓ4 ℓ5 ℓ6 ℓ7
b b b
The Cantor string represented as a fractal harp.
Sean Watson (UCR) Complex Dimensions in MM Spaces 29 / 56
Fractal Strings
Geometric Zeta Function
Given a fractal string L, we define its complex valued geometric zetafunction, ζL as
ζL(s) =∞∑j=1
`sj .
This function is holomorphic on Re s > D =Minkowski dimension of ∂L.Moreover, this half-plane is optimal in the sense that D is the abscissa ofabsolute convergence of ζL.
We define the complex dimensions as the poles of the meromorphiccontinuation.
Sean Watson (UCR) Complex Dimensions in MM Spaces 30 / 56
Fractal Strings
Thus, the Cantor String has geometric zeta function
ζCS(s) =∞∑n=1
2n−1
3ns.
This is absolutely convergent and holomorphic for Re s > log3 2 = D.It has a meromorphic continuation to all of C given by
ζCS(s) =1
3s· 1
1− 2 · 3−s ,
with complex dimensions equal to the set of poles{D + in
2π
log 3: n ∈ Z
}
Sean Watson (UCR) Complex Dimensions in MM Spaces 31 / 56
Fractal Strings
p
0 D 1
The complex dimensions of the ternary Cantor set, with D = log3 2 andp = 2π/ log 3.
Sean Watson (UCR) Complex Dimensions in MM Spaces 32 / 56
Fractal Strings
We find that the Cantor String tube formula (volume of the inner ε-tube)is
VCS(ε) =1
2 log 3
∞∑n=−∞
(2ε)(1−D−inp)
(D + inp)(1− D − inp)− 2ε
where p= 2π/ log 3 and1
2 log 3is the residue of ζCS at each of the poles.
Sean Watson (UCR) Complex Dimensions in MM Spaces 33 / 56
Fractal Strings
Spectrum and Riemann Hypothesis
Given a Minkowski measurable fractal string L, its frequency (spectral)counting function Nν,L(µ) admits a monotonic asymptotic second term ofthe form −cDMµD/2, where D ∈ (0, 1).
The constant cD depends only on D and is directly propotional to −ζ(D),the Riemann zeta function.
This result was obtained by Lapidus and Pomerance, thereby establishing aconnection between the direct spectral problem, Minkowski measurability,and the Riemann zeta function.
Sean Watson (UCR) Complex Dimensions in MM Spaces 34 / 56
Fractal Strings
Inverse Spectral Problem
Given that L is a fractal string for which the spectral counting functionNν,L(µ) admits a monotonic asymptotic second term proportional to µD/2
(as µ→∞), does it follow that L is Minkowski measurable?
It has been shown by Lapidus and Maier that the inverse spectral problemis intimately connected with the location of the critical zeros of ζ(s).
Theorem [LaMa]
The inverse spectral problem has an affirmative answer for all D ∈ (0, 1),with D 6= 1/2, if and only if the Riemann hypothesis is true.
Sean Watson (UCR) Complex Dimensions in MM Spaces 35 / 56
Distance Zeta Function
In order to generalize the theory to RN , the distance zeta function wasintroduced in [FZF].
Distance Zeta Function
Given a bounded set A ⊂ RN , δ > 0, we define the distance zetafunction as
ζA(s) =
∫Aδ
d(x ,A)s−Ndx ,
for s ∈ C with Re s sufficiently large, and the integral taken in theLebesgue sense.
It will turn out that for the properties of this function we wish to study,the value of δ will not matter.
Sean Watson (UCR) Complex Dimensions in MM Spaces 36 / 56
Distance Zeta Function
Theorem [FZF]
Given a bounded set A ⊂ RN , δ > 0, we define the distance zeta function
ζA(s) =
∫Aδ
d(x ,A)s−Ndx .
Then ζA is holomorphic in the half-plane {Re s > dimB(A)} with
ζ ′A(s) =
∫Aδ
d(x ,A)s−N log d(x ,A)dµ.
We have that dimB(A) is optimal, in the sense that it is the abscissa ofLebesgue convergence of ζA.
Further, if D = dimB(A) exists and MD∗ > 0, then ζA(s)→ +∞ as s ∈ R
converges to D from the right.
Sean Watson (UCR) Complex Dimensions in MM Spaces 37 / 56
Distance Zeta Function
If ζA can be meromorphically extended, then we call the poles of such anextension the complex dimensions of the set A.
As we shall see, these poles will differ slightly compared to the geometriczeta function for fractal strings. It is currently unknown if this is perhapsdue to the geometric realization of the fractal strings used.
However, the principal complex dimensions, the poles above the criticalline {Re s = D} where D is the abscissa of holomorphic convergence, willcoincide.
Sean Watson (UCR) Complex Dimensions in MM Spaces 38 / 56
Distance Zeta Function
Given any nontrivial fractal string, L, define A := {ak : k ≥ 1}, whereak :=
∑j≥k lj . Then we have that
ζA(s) = u(s)ζL(s) + v(s)
where u, v are both holomorphic functions in the right half-plane{s ∈ C : Re s > 0}.
Further, the set of poles of the meromorphic extensions of ζA and ζLcoincide to the right of any open half-plane {Re s > c} for c > 0.In particular, the complex dimensions coincide to the right of {Re s = 0}.
Sean Watson (UCR) Complex Dimensions in MM Spaces 39 / 56
Tube Zeta Function
The distance zeta function can be written in the following way: for anyRe s > dimB(A),∫
Aδ
d(x ,A)s−Ndx = δs−N |Aδ|+ (N − s)
∫ δ
0ts−N−1|At |dt.
Tube Zeta Function
Let δ > 0, A a bounded set in RN . Then the tube zeta function of A,ζA, is defined as
ζA(s) =
∫ δ
0ts−N−1|At |dt
Sean Watson (UCR) Complex Dimensions in MM Spaces 40 / 56
Tube Zeta Function
Provided that dimB(A) < N, ζA shares important properties with ζA.
Theorem [FZF]
Assume A is a bounded subset of RN with dimB(A) < N. Then ζA isholomorphic in the half-plane {Re s > dimB(A)}, where dimB(A) is theabscissa of Lebesgue convergence.
Moreover, ζA and ζA will share the same domain of meromorphic extension(if it exists) with the same poles and order.
In particular, the complex dimensions are the same.
Sean Watson (UCR) Complex Dimensions in MM Spaces 41 / 56
Residues and Minkowski Content
Theorem [FZF]
Assume that the bounded set A ⊂ RN is Minkowski nondegenerate (thatis, dimBA = D and 0 <MD
∗ (A) ≤M∗D(A) <∞,) and D < N. If ζA(s)can be meromorphically extended to a neighborhood of s = D, then D isnecessarily a simple pole of ζA(s) and
(N−D)MD∗ (A) ≤ res(ζA(·),D) = (N−D)res(ζA(·),D) ≤ (N−D)M∗D(A).
Sean Watson (UCR) Complex Dimensions in MM Spaces 42 / 56
Residues and Minkowski Content
Let A be the ternary Cantor set, and let δ ≥ 1/6. Then we obtain
ζA(s) = 21−ss−13−s
1− 2 · 3−s + 2δss−1
Its residue at D(A) = log3 2 is equal to
res(ζA(·),D(A)) =2
log 2
(1
6
)log3 2−1=
1
2log3 2 log 2,
while its upper and lower Minkowski contents are given by
Mlog3 2∗ (A) =
log 9
log 3/2
(log 3/2
log 4
)log3 2
, M∗ log3 2(A) = 22−log3 2.
Sean Watson (UCR) Complex Dimensions in MM Spaces 43 / 56
Residues and Minkowski Content
More generally, at each of the poles on the critical line {Res = log3 2},sk := log3 2 + kpi , k ∈ Z,with p :=
2π
log 3, we have
res(ζA(·), sk) =log3 2
sk2kpires(ζA(·),D(A)).
In particular, it is noteworthy that these residues tend to zero as k → ±∞.
Sean Watson (UCR) Complex Dimensions in MM Spaces 44 / 56
Sierpinski carpet
In parallel to the fractal string case, we will call V (t) := |At | the volumeof the t-tube neighborhood.
Let A be the standard Sierpinski carpet in the plane. Then
|At | = t2−D(G (log t−1) + O(tD−1))
as t → 0, where D = log3 8 and G is a nonconstant periodic function withperiod T = log 3.
By direct computation we can find that both zeta functions have ameromorphic extension to all of C, and the set of complex dimensions of Aare simple poles given by
dimCA =
{D +
2π
log 3ki : k ∈ Z
}.
Sean Watson (UCR) Complex Dimensions in MM Spaces 45 / 56
Sierpinski carpet
Sean Watson (UCR) Complex Dimensions in MM Spaces 46 / 56
Sierpinski carpet
1 2
p
0 D
The complex dimensions of the Sierpinski carpet, with D = log3 8 andp = 2π/ log 3.
Sean Watson (UCR) Complex Dimensions in MM Spaces 47 / 56
Sphere in RN
Let BR(0) be the open ball in Rn with radius R, and let A = ∂BR(0) bethe (N − 1)-dimensional sphere with radius R. Further, let ck = 1− (−1)k
and ωN = |B1(0)|.
Fix δ < R. Then ζA(s) meromorphically extends to all C, withrepresentation
ζA(s) = ωN
N∑k=0
ckRn−k(N
k
)δs−N+k
s − (N − k).
As expected, we get D(A) = N − 1, although (perhaps surprisingly) its setof complex dimensions is
dimC(A) =
{N − 1,N − 3, . . . ,N − (2
⌊N − 1
2
⌋+ 1)
}.
Sean Watson (UCR) Complex Dimensions in MM Spaces 48 / 56
Fractality
The introduction of complex dimensions has led to a conjectured definitionof fractality, first in [FGCD] and then further extended in [FZF], thatwould not leave out exceptional cases:
Definition
A geometric object is said to be a fractal subset of RN if its associatedfractal zeta function has at least one nonreal complex dimension.
This would include not only classical fractals, but former exceptions suchas the Devil’s staircase.
Sean Watson (UCR) Complex Dimensions in MM Spaces 49 / 56
Generalized Minkowski Dimension
Minkowski Dimension in MM spaces
Let X be an Ahlfors regular MM space of dimension DH . Then we definethe r-dimensional upper Minkowski content by
M∗r = lim supt→0
|At |tDH−r
.
Lower content and Minkowski dimension are then defined analogously tothe Euclidean case.
Ahlfors regularity is necessary to insure that the ”ambient space” hasconsistent dimension, or generally that single points have Minkowskidimension 0. See weighted RN space at A = {0} for a counterexample.
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Preliminary Results
Theorem (Lapidus, W.)
If we define D(A) = dimBA, then the distance zeta function,
ζA(s) =
∫Aδ
d(x ,A)s−DH dµ,
is holomorphic in the half plane {Re s > D(A)}, with
ζ ′A(s) =
∫Aδ
d(x ,A)s−DH log d(x ,A)dµ.
We have that D(A) is optimal, in the sense that it is the abscissa ofLebesgue convergence of ζA.
Further, if D = dimB(A) exists and MD∗ > 0, then ζA(s)→ +∞ as s ∈ R
converges to D from the right.
Sean Watson (UCR) Complex Dimensions in MM Spaces 51 / 56
Future Work
Is the analogous tube zeta function well defined and do further resultsinvolving these fractal zeta functions (relative fractal zeta functions,spectrums, etc.) generalize to MM spaces?
Find and study examples of fractal sets (sets with nonreal complexdimensions) in MM spaces.
Examine what information the complex dimensions give concerningthe geometry of such fractal sets, in analogy to the Euclidean case.
Sean Watson (UCR) Complex Dimensions in MM Spaces 52 / 56
Bibliography I
[1] A. Bjorn, J. Bjorn
Nonlinear Potential Theory on Metric Spaces
European Mathematical Society, 2011.
[2] J. Cheeger, B. Kleiner
Realization of Metric Spaces as Inverse Limits and Bilipschitz Embdding in L1
Geom. Funct.Anal., (23) 1 (2013), 96-133.
[3] R. Coifman, G. Weiss
Analyse harmonique non-commutative sur certains espaces homogenes
Lecture Notes in Math., vol. 242, Springer-Verlag, Berlin and New York, 1971.
[4] G. Dafni, R. J. McCann, A. Stancu (eds.)
Analysis and Geometry of Metric Measure Spaces, Lecture Notes of the 50thSeminaire de Mathematiques Superieures (SMS), (Montreal 2011)
CRM Proceedings & Lecture Notes, 56, Centre de Recherches Mathematiques(CRM), Montreal and Amer. Math. Soc., Providence, R.I., 2013.
Sean Watson (UCR) Complex Dimensions in MM Spaces 53 / 56
Bibliography II
[5] D. Danielli, N Garogalo, D. Nhieu
Non-Doubling Ahlfors Measures, Perimeter Measures, and the Characterization ofthe Trace Spaces of Sobolev Functions in Carnot-Caratheodory Spaces
Mem. Am. Math. Soc., (857) 182 (2006), 1-119.
[6] G. David, S. Semmes
Fractured Fractals and Broken Dreams: Self-Similar Geometry through Metric andMeasure
Oxford University Press, Oxford, 1997.
[7] J. Heinonen
Nonsmooth calculus
Bull. Amer. Math. Soc., 44 (2007), 163-232.
[8] N. Kajino
Analysis and geometry of the measurable Riemannian structure on the Sierpinskigasket
Contemp. Math., 600 (2013), 91-134.
Sean Watson (UCR) Complex Dimensions in MM Spaces 54 / 56
Bibliography III
[9] J. Kigami
Volume Doubling Measures and Heat Kernel Estimates of Self-Similar Sets
Mem. Amer. Math. Soc., (932) 199 (2009), 94 pages.
[10] M.L. Lapidus, M. Frankenhuijsen
Fractal Geometry, Complex Dimensions and Zeta Functions: Geometry and Spectraof Fractal Strings, second edition (of the 2006 edition)
Springer Monograph in Mathematics, Springer, New York, 2013, 593 pages.
[11] M.L. Lapidus, H. Maier
The Riemann hypothesis and inverse spectral problems for fractal strings
J. London Math. Soc. (2) 52 (1995), 15-34.
[12] M.L. Lapidus, G. Radunovic, D. Zubrinic
Fractal Zeta Functions and Fractal Drums: Higher-Dimensional Theory of ComplexDimensions
research monograph, preprint, 317 pages. Expected submission date: June 2014.
Sean Watson (UCR) Complex Dimensions in MM Spaces 55 / 56
Bibliography IV
[13] M.L. Lapidus, G. Radunovic, D. Zubrinic
Distance and Tube zeta functions of arbitrary compact sets and relative fractaldrums in Euclidean spaces
article in preparation, 2014.
[14] M.L. Lapidus, G. Radunovic, D. Zubrinic
Meromorphic extensions of fractal zeta functions
article in preparation, 2014.
[15] M.L. Lapidus, G. Radunovic, D. Zubrinic
Fractal zeta functions, complex dimensions and relative fractal drums
survey article in preparation, 2014.
[16] M.L. Lapidus, J. Sarhad
Dirac operators and geodesic metric on the harmonic Sierpinski gasket and otherfractal sets
To appear in J. Noncommutative Geometry, 2014. arXiv:1212.0878 [math.MG],Feb. 2014.
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